The first coordinate of a point that is chosen at random in the Euclidean ball of radius $\sqrt{3} $ according to the uniform distribution. Calculate the distribution function, mean and variance of $x_{1}$.
My approach:
I think that $x_{1}$ most be in $[-\sqrt{3}, \sqrt{3}]$. If that is the case I have the following function: \begin{align*} F_{x_{1}}=P[x_{1}\leq y] = \left\{ \begin{array}{cc} 0 & \text{if}\ \ y<-\sqrt{3}\\[2em] 1 & \text{if}\ \ y<\sqrt{3}\\ ? & \text{if}\ \ -\sqrt{3}<y<\sqrt{3} \end{array} \right.\text{for every}\ y\in\mathbb{R} \end{align*}
How do I find the last part of the preceding function?. I know I must find an integral. Assuming I have X, Y and Z axes, should I do a triple integral considering the $dxdydz$?